In a recent paper entitled "High temperature expansion applied to fermions near Feshbach resonance" (Phys. Rev. Lett. 92, 160404 (2004)), Ho and Muller have demonstrated a remarkable similarity between its high and low temperature properties at resonance. The quantum second virial coefficient plays a crucial role in their analysis, and has a universal value at resonance. In this paper, we explore the connection between the quantum and classical second virial coefficients, and show that near a Feshbach resonance, an exact mapping from the quantum to classical form is possible. This gives rise to a scale independent inverse square potential for the classical second virial coefficient. It is suggested that it may be tested by measuring the isothermal compressibility of the gas on the repulsive side of the resonance.
We use the virial expansion to investigate the behavior of the two-component, attractive Fermi gas in the high-temperature limit, where the system smoothly evolves from weakly attractive fermions to weakly repulsive bosonic dimers as the short-range attraction is increased. We present a formalism for computing the virial coefficients that employs a diagrammatic approach to the grand potential and allows one to easily include an effective range
R∗ in the interaction. In the limit where the thermal wavelength $\ensuremath{\lambda}\ensuremath{\ll}{R}^{*}$, the calculation of the virial coefficients is perturbative even at unitarity and the system resembles a weakly interacting Bose-Fermi mixture for all scattering lengths
a. By interpolating from the perturbative limits $\ensuremath{\lambda}/|a|\ensuremath{\gg}1$ and ${R}^{*}/\ensuremath{\lambda}\ensuremath{\gg}1$, we estimate the value of the fourth virial coefficient at unitarity for
R∗=0 and we find that it is close to the value obtained in recent experiments. We also derive the equations of state for the pressure, density, and entropy, as well as the spectral function at high temperatures.
The quantum theory of the third virial coefficient
C is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of
CBE are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for
He4 and
He3, by means of a comparison of the corresponding expansion for the second virial coefficient
B with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to
B. Then the leading term of
C is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for
C is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.
The virial expansion method is applied within a harmonic approximation to an interacting N-body system of identical fermions. We compute the canonical partition functions for two and three particles to get the two lowest orders in the expansion. The energy spectrum is carefully interpolated to reproduce ground-state properties at low temperature and the noninteracting high-temperature limit of constant virial coefficients. This resembles the smearing of shell effects in finite systems with increasing temperature. Numerical results are discussed for the second and third virial coefficients as functions of dimension, temperature, interaction, and transition temperature between low- and high-energy limits.
We theoretically investigate the high-temperature thermodynamics of a strongly interacting trapped Fermi gas near either
s-wave or
p-wave Feshbach resonances, using a second-order quantum virial expansion. The second virial coefficient is calculated based on the energy spectrum of two interacting fermions in a harmonic trap. We consider both isotropic and anisotropic harmonic potentials. For the two-fermion interaction, either
s-wave or
p-wave, we use a pseudopotential parametrized by a scattering length and an effective range. This turns out to be the simplest way of encoding the energy dependence of the low-energy scattering amplitude or phase shift. This treatment of the pseudopotential can be easily generalized to higher partial-wave interactions. We discuss how the second virial coefficient and thermodynamics are affected by the existence of these finite-range interaction effects. The virial expansion result for a strongly interacting
s-wave Fermi gas has already been proved very useful. In the case of
p-wave interactions, our results for the high-temperature equation of state are applicable to future high-precision thermodynamic measurements for a spin-polarized Fermi gas near a
p-wave Feshbach resonance.